A sharp compactness theorem for genus-one pseudo-holomorphic maps

Abstract

For every compact almost Kahler manifold $\left(X,\omega ,J\right)$ and an integral second homology class $A$, we describe a natural closed subspace ${\overline{\mathfrak{M}}}_{1,k}^0\left(X,A;J\right)$ of the moduli space ${\overline{\mathfrak{M}}}_{1,k}\left(X,A;J\right)$ of stable $J$–holomorphic genus-one maps such that ${\overline{\mathfrak{M}}}_{1,k}^0\left(X,A;J\right)$ contains all stable maps with smooth domains. If $\left({\mathbb{P}}^n,\omega ,J_0\right)$ is the standard complex projective space, ${\overline{\mathfrak{M}}}_{1,k}^0\left({\mathbb{P}}^n,A;J_0\right)$ is an irreducible component of ${\overline{\mathfrak{M}}}_{1,k}\left({\mathbb{P}}^n,A;J_0\right)$. We also show that if an almost complex structure $J$ on ${\mathbb{P}}^n$ is sufficiently close to $J_0$, the structure of the space ${\overline{\mathfrak{M}}}_{1,k}^0\left({\mathbb{P}}^n,A;J\right)$ is similar to that of ${\overline{\mathfrak{M}}}_{1,k}^0\left({\mathbb{P}}^n,A;J_0\right)$. This paper’s compactness and structure theorems lead to new invariants for some symplectic manifolds, which are generalized to arbitrary symplectic manifolds in a separate paper. Relatedly, the smaller moduli space ${\overline{\mathfrak{M}}}_{1,k}^0\left(X,A;J\right)$ is useful for computing the genus-one Gromov–Witten invariants, which arise from the larger moduli space ${\overline{\mathfrak{M}}}_{1,k}\left(X,A;J\right)$.